INSTRUCTOR: Let v equal the vector negative 1, comma, negative 1, comma, 1 and the vector w be equal to the vector 2, comma, 0, comma, 1. Find a unit vector in the direction of 5v plus 3w. So I'm going to call this vector u. Vector u equals 5v plus 3w.

This will be equal to 5 multiplied by that component form negative 1, comma, negative 1, comma, 1. Plus 3 of the vector 2, comma, 0, comma, 1. And we will multiply by the scalar, the 5 and the 3, distributing those to each component. So this is the vector negative 5, comma, negative 5, comma, 5 plus the vector 6, comma, 0, comma, 3.

Again, we're multiplying by that scalar, and then we'll add these component y's. So we'll have negative 5 plus 6. It makes our x component 1. Our y component will be negative 5 plus 0, which would be negative 5. And our z component will be 5 plus 3, or 8. All right. So the vector that we were talking about has the direction of 5v plus 3w is 1, comma, negative 5, comma, 8.

However, this must be a unit vector. So let's find the magnitude of this vector, magnitude of u. And that will be the square root of 1 squared plus negative 5 squared plus 8 squared. Now, 1 squared is 1. Negative 5 squared is 25. 8 squared is 64.

So that this is equal to the square root of 90, which simplifies as 3 square roots of 10. And you may notice, this is not a unit vector. This is not a unit vector. So the vector we are actually looking for is going to be that vector u, but we're going to divide by the magnitude of that vector. So vector u divided by the magnitude of vector u.

And that means we're going to have the vector 1 divided by 3 square roots of 10, comma, negative 5 divided by 3 square roots of 10, comma, 8 divided by 3 square roots of 10. Now, we know that this is, in fact, in that same direction because of the work we've done. And we now know that it is a unit vector, because we've divided by its magnitude so that the magnitude of this must be 1.